Approximative shape. I: Basic notions
Resolutions of spaces and maps [introduced by \textit{S. Mardešić} in Fund. Math. 114, 53-78 (1981; Zbl 0411.54019)] are important tools in shape theory. A problem, however, is that a map \(f: X\to Y\) in general fails to have a resolution \(f: {\mathcal X}\to {\mathcal Y}\) with respect to arbitrary polyhedral resolutions \(p: X\to {\mathcal X}\) and \(q: Y\to {\mathcal Y}\). The present paper solves this problem by introducing the concept of approximative resolutions. Let C be a subcategory of Top, e.g. the full subcategory AP of approximate polyhedra. The approximative pro-category Appro-C is defined as follows. The objects are approximative inverse systems (\({\mathcal X},{\mathcal U})\) over C consisting of a cofinite inverse system \({\mathcal X}=\{X_{\alpha}\}\) over C plus a system \({\mathcal U}=\{U_{\alpha}\}\) of normal open coverings \(U_{\alpha}\) of \(X_{\alpha}\) such that certain refinement conditions are fulfilled. The morphisms are suitable equivalence classes of approximative system maps (the latter differ from maps of inverse systems in replacing the standard commutativity conditions by closeness conditions with respect to the coverings). An approximative C-resolution of a space X is an object (\({\mathcal X},{\mathcal U})\) of Appro-C together with an ordinary resolution \(p: X\to {\mathcal X}\) (for \(C=Top\), one speaks about approximative resolutions). An approximative resolution of a map \(f: X\to Y\) with respect to approximative resolutions \(p: X\to ({\mathcal X},{\mathcal U})\) and \(q: Y\to ({\mathcal Y},{\mathcal V})\) is an approximative system map \(f: ({\mathcal X},{\mathcal U})\to ({\mathcal Y},{\mathcal V})\) such that \(qf\) and \(fp\) are \({\mathcal V}\)-close. One of the main results is that for any map \(f: X\to Y\), any approximative resolution \(p: X\to ({\mathcal X},{\mathcal U})\) and any approximative AP-resolution \(q: Y\to ({\mathcal Y},{\mathcal V})\), there exists an approximative resolution \(f: ({\mathcal X},{\mathcal U})\to ({\mathcal Y},{\mathcal V})\) of f with respect to \(p\) and \(q\). The author then defines the approximative shape category ASh. Its objects are all topological spaces, and its morphisms \(X\to Y\) are suitable equivalence classes of morphisms (\({\mathcal X},{\mathcal U})\to ({\mathcal Y},{\mathcal V})\), where \(p: X\to ({\mathcal X},{\mathcal U})\) and \(q: Y\to ({\mathcal Y},{\mathcal V})\) are approximative AP- resolutions. This construction does not involve homotopies: On the level of Top, it is the precise analogue of the Mardešić-Segal construction of the shape category which starts with the homotopy category HTop and uses HPol-extensions of spaces. The paper also investigates Morita's Tychonoff functor, the completion functor and the so-called realization functor on ASh. The realization functor is shown to establish a category isomorphism from the full subcategory ASh(CTop 3.5) of ASh to the full subcategory CTop 3.5 of Top whose objects are the topologically complete Tychonoff spaces.
scientific article; zbMATH DE number 4054600